11.2.2.2 - Example: Summarized Data, Different Proportions

Example: Roulette Section

An American roulette wheel contains 38 slots: 18 red, 18 black, and 2 green. A casino has purchased a new wheel and they want to know if there is any evidence that the wheel is unfair. They spin the wheel 100 times and it lands on red 44 times, black 49 times, and green 7 times.

Use Minitab to conduct a hypothesis test to address this question. 

We'll go through each of the steps in the hypotheses test:

Step 1: Check assumptions and write hypotheses

If the wheel is 'fair' then the probability of red and black are both 18/38 and the probability of green is 2/38.

\(H_0\colon p_{red}=\dfrac{18}{38}, p_{black}=\dfrac{18}{38}, p_{green}=\dfrac{2}{38}\)
\(H_a\colon\) at least one \(p_i\) is not as specified in the null

We can use the null hypothesis to check the assumption that all expected counts are at least 5.

\(Expected\;count=n (p_i)\)

With n = 100 we meet the assumptions needed to use Chi-square.

Step 2: Compute the test statistic

Let's use Minitab to calculate this.

First, enter the summarized data into a Minitab Worksheet.

  C1 C2
  Color Count
1 Red 44
2 Black 49
3 Green 7
  1. After entering the data, select Stat > Tables > Chi-Square Goodness of Fit Test (One Variable)
  2. Double-click Count to enter it into the Observed Counts box
  3. Double-click Color to enter it into the Category names (optional) box
  4. For Test select Input constants
  5. Select Proportions specified by historical counts (this is what we would expect if the null was true)
  6. Enter 18/38 for Black, 2/38 for Green and 18/38 for Red
  7. Click OK

This should result in the following output:

Chi-Square Goodness-of-Fit Test: Count

Observed and Expected Counts
Category Observed Historical Counts Test
Proportion
Expected Contribution
to Chi-Sq
Red 44 18 0.473684 47.3684 0.239532
Black 49 18 0.473684 47.3684 0.056199
Green 7 2 0.052632 5.2632 0.573158
Chi-Square Test
N DF Chi-Sq P-Value
100 2 0.868889 0.648

The test statistic is a Chi-Square of 0.87.

Step 3: Determine the p-value
The p-value from the output is 0.648.
 
Step 4: Make a decision

\(p>0.05\) therefore we fail to reject the null hypothesis.

Step 5: State a "real world" conclusion

There is not evidence that this roulette wheel is unfair.